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\title[Divide and Conquer Algorithm in CEM]{分治混合算法在计算电磁学中的应用}
\subtitle{
  Applications of Divide and Conquer Algorithm\\
  in Computational Electromagnetics
}

\author[Li You]{李友}
\institute[CCEM, at SEU]{
  东南大学计算电磁学研究中心\\
  Center for Computational Electromagnetics\\
  Southeast University
}
\date[\today]{\today}
%\beamertemplatetransparentcovereddynamic
\AtBeginSection[]{
  \frame<handout:0>{
    \frametitle{Outline}
    \tableofcontents[current,currentsubsection]
  }
}

\begin{document}
\frame[plain]{\titlepage}

\section{PO-MM Hybrid}
\frame{
\frametitle{Physical Optics}
\pause
在高频条件下，可以采用散射体表面的感应电流取代散射体本身作为散射场的源，然后对表面感应电流积分而求得散射场。
\pause
\begin{columns}
\begin{column}{.6\textwidth}
在PEC表面上，根据边界条件有
\[\bar{M}^{sca}=-\hat{n}\times\bar{E}=0\]
\[\bar{J}^{sca}=\hat{n}\times\bar{H}\]
\pause
由于PO基于无限大切平面假设，应用镜像原理可得
\begin{theorem}
\[\bar{J}^{sca}=\hat{n}\times\bar{H}=2\hat{n}\times\bar{H}^{inc}\]
\end{theorem}
\pause
因此PO区的感应电流强度可直接由入射磁场的强度得出，无需求解线性方程组。
\end{column}
\pause
\begin{column}{.4\textwidth}
\begin{figure}[htbp]
\includegraphics[scale=0.15]{../data/chapter3/5mPlate.jpg}
\end{figure}
\pause
\begin{figure}[htbp]
\includegraphics[scale=0.25]{../data/chapter3/POvsMLFMA.pdf}
\end{figure}
\end{column}
\end{columns}
}

\frame{
\frametitle{Problems in PO-Only}
\pause
\begin{columns}
\begin{column}{.5\textwidth}
\begin{figure}[htbp]
\includegraphics[scale=0.2]{../data/chapter3/sphere_plate_30m.jpg}
\end{figure}
在散射问题中，电小尺寸或复杂结构上高频方法计算精度太低。
\end{column}
\pause
\begin{column}{.5\textwidth}
\begin{figure}[htbp]
\includegraphics[scale=0.5]{../diaplo_sphere.eps}
\end{figure}
\[\bar{J}^{sca}=\hat{n}\times\bar{H}=2\hat{n}\times\bar{H}^{inc}\]
在辐射问题中，球体表面的$\bar{H}^{inc}$为零，意味着没有PO电流的产生，这显然是不正确的。
\end{column}
\end{columns}
}

\frame{
\frametitle{Hybrid PO with MoM}
\pause
因此这里引入修正后的PO区电流表达式
\[\begin{split}
\bar{J}^{PO}\left(\bar{r}\right)&=2\delta_i\cdot\hat{n}\times\bar{H}^{inc}\left(\bar{r}\right)\\
&+\sum_{n=1}^{N_J^{MM}}2\alpha_n\delta_{J,n}\cdot\hat{n}\times\bar{\mathcal{L}_J^H}\bar{f}_n\\
&+\sum_{n=1}^{N_I^{MM}}2\beta_n\delta_{I,n}\cdot\hat{n}\times\bar{\mathcal{L}_I^H}g_n
\end{split}\]
}

\frame{
\frametitle{Key Technology in PO-MM}
\pause
为了简化上式中算子$\bar{\mathcal{L}}$的计算，将PO电流同样用RWG基函数展开。
\begin{columns}
\begin{column}{.5\textwidth}
\begin{figure}[htbp]
\includegraphics[scale=0.5]{../PO_RWG.eps}
\end{figure}
\end{column}
\begin{column}{.5\textwidth}
\[\bar{J}^{PO}=\sum_{n=1}^{N_J^{PO}}\gamma_n\cdot\bar{f}_n\]
\[\gamma_n=\frac{1}{2}\left(\hat{t}_n^++\hat{t}_n^-\right)\cdot\bar{J}^{PO}\left(\bar{r}_n\right)\]
\end{column}
\end{columns}
\pause
最后将积分方程写成下列形式
\[\left(\bar{\mathcal{L}}_J^E\bar{J}^{MM}\right)_{tan}+
\left(\bar{\mathcal{L}}_I^EI^{MM}\right)_{tan}+
\left(\bar{\mathcal{L}}_J^E\bar{J}^{PO}\right)_{tan}=-\bar{E}_{tan}^{inc}\]
}

\frame{
\frametitle{Examples-Radiation Problems}
\pause
\begin{columns}
\begin{column}{.3\textwidth}
\begin{figure}[htbp]
  \includegraphics[scale=0.15]{../data/chapter3/dipole_plate.jpg}
\end{figure}
\end{column}
\begin{column}{.3\textwidth}
  \begin{figure}[htbp]
    \includegraphics[scale=0.25]{../data/chapter3/dipole_plate_V.pdf}
  \end{figure}
\end{column}
\begin{column}{.3\textwidth}
  \begin{figure}[htbp]
    \includegraphics[scale=0.25]{../data/chapter3/dipole_plate_H.pdf}
  \end{figure}
\end{column}
\end{columns}
\pause
\begin{columns}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.2]{../data/chapter3/dipole_sphere.jpg}
    \end{figure}
  \end{column}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter3/dipole_sphere.pdf}
    \end{figure}
  \end{column}
\end{columns}
}

\frame{
\frametitle{Examples-Scattering Problems}
\pause
\begin{columns}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.2]{../data/chapter3/sphere_plate.jpg}
    \end{figure}
  \end{column}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter3/sphere_plate.pdf}
    \end{figure}
  \end{column}
\end{columns}
\pause
\begin{columns}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.2]{../data/chapter3/f16_sphere.pdf}
    \end{figure}
  \end{column}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter3/f16_sphere_PO_MoM_MLFMA_VV.pdf}
    \end{figure}
  \end{column}
\end{columns}
}

\frame{
\frametitle{Advantages}
\pause
\begin{table}[htbp]
\caption{使用混合方法前后未知量数目的对比}
\centering
\begin{tabular}{llll}
\hline
算例 & 单一矩量法 & 使用PO-MM混合 & 未知数减少量\\
\hline
线天线+平板 &  5011 & 9 & 99.82\%\\
线天线+球 & 3011 & 9 & 99.70\%\\
球+平板 & 11288 & 3072 & 72.79\%\\
球+飞机 & 121843 & 3072 & 97.48\%\\
\hline
\end{tabular}
\end{table}
}

\section{DDM in MoM Method}
\frame{
\frametitle{Domain Decomposition Method}
\pause
将原始求解$Ax=b$的问题分解到两个子域
\[\left[\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}\\
\end{array}\right]
\left[\begin{array}{c}
x_1\\
x_2\\
\end{array}\right]
=\left[\begin{array}{c}
b_1\\
b_2\\
\end{array}\right]
\]
\pause

为了并行求解上述方程，常用一种Jacobi型区域分解法
\[\left[\begin{array}{cc}
A_{11} & A_{12}\\
A_{21} & A_{22}\\
\end{array}\right]
=\left[\begin{array}{cc}
A_{11} & 0\\
0 & A_{22}\\
\end{array}\right]
+\left[\begin{array}{cc}
0 & A_{12}\\
A_{21} & 0\\
\end{array}\right]
\]
}

\frame{
\frametitle{Domain Decomposition Method}
\pause
原方程的迭代解法可变为
\[\left[\begin{array}{cc}
A_{11} & 0\\
0 & A_{22}\\
\end{array}\right]
\left[\begin{array}{c}
x_1\\
x_2\\
\end{array}\right]^{(n)}
=\left[\begin{array}{c}
b_1\\
b_2\\
\end{array}\right]
-\left[\begin{array}{cc}
0 & A_{12}\\
A_{21} & 0\\
\end{array}\right]
\left[\begin{array}{c}
x_1\\
x_2\\
\end{array}\right]^{(n-1)}
\]
\pause

上式可转换为两个耦合的系统
\[\begin{split}
&{A_{11}x_1}^{(n)}=b_1-{A_{12}x_2}^{(n-1)}\\
&{A_{22}x_2}^{(n)}=b_2-{A_{21}x_1}^{(n-1)}
\end{split}
\]
}

\frame{
\frametitle{Partition Scheme}
\pause
\begin{columns}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.3]{../DDM_partition.eps}
    \end{figure}
  \end{column}
\pause

  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.5]{../buffer_region.jpg}
    \end{figure}
  \end{column}
\end{columns}
\pause


因此原算法变为：每个子区域$\Omega_i$用其附近的缓冲区$\Omega_{bi}$进行扩展，依次对扩展后的区域$\Omega_i'=\Omega_i+\Omega_{bi}$求解，舍弃缓冲区的电流，保留并更新原子区域$\Omega_i$的电流，这里定义各扩展子区域的补余区域$\bar{\Omega_i}=\Omega-\Omega_i'$。
}

\frame{
\frametitle{Iteration Equations}
\pause
根据上述原理，可建立子区域间的EFIE积分公式
\[\bar{K}_e\left(\bar{r},\bar{J}\right)=
jk\eta\left[\int_{\Omega_i}ds'\bar{G}\left(\bar{r},\bar{r}'\right)\cdot\bar{J}\left(\bar{r}'\right)\right]_{tan},\bar{r}'\in\bar{\Omega_i},\bar{r}\in\Omega_i'
\]
\[
\bar{T}_e\left(\bar{r},\bar{J}\right)=
jk\eta\left[\int_{\Omega_i'}ds'\bar{G}\left(\bar{r},\bar{r}'\right)\cdot\bar{J}\left(\bar{r}'\right)\right]_{tan},\bar{r}'\in\Omega_i',\bar{r}\in\Omega_i'
\]
\pause
其中，$\bar{G}$为自由空间内的并矢格林函数
\[\bar{G}=\left[\bar{I}-\frac{1}{k^2}\nabla\nabla'\right]G\left(\bar{r},\bar{r}'\right)=\left[\bar{I}-\frac{1}{k^2}\nabla\nabla\right]\frac{e^{-jk|\bar{r}-\bar{r}'|}}{|\bar{r}-\bar{r}'|}
\]
\pause
子区域间的迭代公式为
\[\bar{T}_e\left(\bar{r},\bar{J}^{\left(k\right)}\right)=
-\bar{K}_e\left(\bar{r},\bar{J}^{\left(k-1\right)}\right)
+\left[\bar{E}^{inc}\left(\bar{r}\right)\right]_{tan},\bar{r}\in\Omega_i'
\]
}

\frame{
\frametitle{Iterative Equations}
\pause
从数学角度来看，上述区域分解的迭代算法可以表示为矩阵公式
\[\tilde{Z}_{ii}\tilde{I}_i^{(k)}=\tilde{V}_i-\sum_{j<i,c(j)\notin b(i)}\tilde{Z}_{ij}I_j^{(k)}-\sum_{j>i,c(j)\notin b(i)}\tilde{Z}_{ij}I_j^{(k-1)}
\]
\pause
$\tilde{Z}_{ii}$与$\tilde{Z}_{ij}$由$\Omega_i$和缓冲区$\Omega_{bi}$的基函数，$\Omega_j$的基函数相互作用生成，分别为
\[\tilde{Z}_{ii}=\left[
\begin{array}{cc}
Z_{ii} & Z_{ib(i)}\\
Z_{b(i)i} & Z_{b(i)b(i)}
\end{array}
\right],\quad
\tilde{Z}_{ij}=\left[
\begin{array}{c}
Z_{ij}\\
Z_{b(i)j}
\end{array}
\right]
\]
\pause
$I_j^{(k)}$是$\Omega_j$上第$k$次迭代求解的电流，$\tilde{I}_i^{(k)}$是$\Omega_i'$上第$k$次迭代求解的电流，$\tilde{V}_i$是$\Omega_i'$上的入射场,它们分别表示为
\[\tilde{I}_i^{(k)}=\left[
\begin{array}{c}
I_i^{(k)}\\
I_{b(i)}
\end{array}
\right],\quad
\tilde{V}_i=\left[
\begin{array}{c}
V_i\\
V_{b(i)}
\end{array}
\right]
\]
}

\frame{
\frametitle{Examples-Open Structure}
\pause
\begin{columns}
  \begin{column}{.3\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter4/plate.jpg}
    \end{figure}
    \begin{figure}[htbp]
      \includegraphics[scale=0.2]{../data/chapter4/plate.pdf}
    \end{figure}
  \end{column}
\pause
  \begin{column}{.3\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter4/disk.jpg}
    \end{figure}
    \begin{figure}[htbp]
      \includegraphics[scale=0.2]{../data/chapter4/disk.pdf}
    \end{figure}
  \end{column}
\pause
  \begin{column}{.3\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter4/box.jpg}
    \end{figure}
    \begin{figure}[htbp]
      \includegraphics[scale=0.2]{../data/chapter4/box.pdf}
    \end{figure}
  \end{column}
\end{columns}
}

\frame{
\frametitle{Examples-Closed Structure}
\pause
\begin{columns}
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter4/pyramid.jpg}
    \end{figure}
    \begin{figure}[htbp]
      \includegraphics[scale=0.3]{../data/chapter4/pyramid.pdf}
    \end{figure}
  \end{column}
\pause
  \begin{column}{.5\textwidth}
    \begin{figure}[htbp]
      \includegraphics[scale=0.25]{../data/chapter4/cylinder.jpg}
    \end{figure}
    \begin{figure}[htbp]
      \includegraphics[scale=0.3]{../data/chapter4/cylinder.pdf}
    \end{figure}
  \end{column}
\end{columns}
}

\frame{
\frametitle{Efficiency}
\pause
\begin{table}[htbp]
\caption{DDM与普通矩量法的内存效率对比(MB)}
%\label{DDM_memory}
\centering
\begin{tabular}{llllll}
\hline
算法 & 平板 & 圆盘 & 盒子 & 金字塔 & 圆柱\\
\hline
MoM & 54 & 14 & 35 & 15 & 126\\
DDM & 23 & 6 & 12 & 12.5 & 23\\
内存减少量 & 57.4\% & 57.1\% & 65.7\% & 20\% & 81.7\%\\
\hline
\end{tabular}
\end{table}
\pause
\begin{table}[htbp]
\caption{DDM与普通矩量法的时间效率对比(s)}
\centering
\begin{tabular}{llllll}
\hline
算法 & 平板 & 圆盘 & 盒子 & 金字塔 & 圆柱\\
\hline
MoM & 32.146 & 8.907 & 23.057 & 10.434 & 74.841\\
DDM & 12.917 & 3.272 & 16.825 & 27.521 & 72.949\\
时间减少量 & 59.8\% & 63.3\% & 27.0\% & -163.8\% & 2.5\%\\ 
\hline
\end{tabular}
\end{table}
}

\section{MoM based EPA}
\frame{
\frametitle{Surface Integral Equation}
\pause
基于表面积分方程问题的散射场可以通过下列形式得到
\[\bar{E}_p^{sca}\left(\bar{r}\right)=\eta_0\bar{\mathcal{L}}_p\left(\bar{J}_p\right)\left(\bar{r}\right)
-\bar{\mathcal{K}}_p\left(\bar{M}_p\right)\left(\bar{r}\right)\]
\[\bar{H}_p^{sca}\left(\bar{r}\right)=\bar{\mathcal{L}}_p\left(\bar{M}_p\right)/\eta_0
+\bar{\mathcal{K}}_p\left(\bar{J}_p\right)\left(\bar{r}\right)\]
\pause
写成矩阵形式
\[\left[\begin{array}{c}
\bar{E}_p^{sca}\\
\eta_0\bar{H}_p^{sca}
\end{array}\right]=
\left[\begin{array}{cc}
\eta_0\bar{\mathcal{L}}_p & -\bar{\mathcal{K}}_p\\
\eta_0\bar{\mathcal{K}}_p & \bar{\mathcal{L}}_p
\end{array}\right]
\left[\begin{array}{c}
\bar{J}_p\\
\bar{M}_p
\end{array}\right]\]
\pause
\begin{theorem}
  处于封闭区域内部或外部的场可以通过区域边界上场的切向分量决定。
\end{theorem}
}

\frame{
\frametitle{Equivalence Principle Algorithm}
\pause
EPA的基本思想是将原物体表面的未知量等效转移到空间的虚拟表面上，使得虚拟面外部区域的散射场与原问题一致。
\pause
\begin{figure}[htbp]
  \centering
  \includegraphics[scale=0.14]{../EPA.eps}
\end{figure}
}

\frame{
\frametitle{Equivalence Principle Algorithm}
\pause
在虚拟面上有新方程
\[\left[\begin{array}{c}
\bar{J}_p^{sca}\\
\bar{M}_p^{sca}
\end{array}\right]=
S_{pp}\left[\begin{array}{c}
\bar{J}_p^{inc}\\
\bar{M}_p^{inc}
\end{array}\right]+
S_{pp}\left(\sum_{l=1,l\ne p}^PT_{pl}\left[\begin{array}{c}
\bar{J}_l^{sca}\\
\bar{M}_l^{sca}
\end{array}\right]\right)\]
\pause
其中
\[S_{pp}=C_p\left(B_{pp}\right)^{-1}A_p\]
\pause
\[T_{pl}=\left[\begin{array}{cc}
\hat{n}_p\times\bar{\mathcal{K}}_{pl} & \hat{n}_p\times\bar{\mathcal{L}}_{pl}/\eta_0\\
-\eta_0\hat{n}_p\times\bar{\mathcal{L}}_{pl} & \hat{n}_p\times\bar{\mathcal{K}}_{pl}
\end{array}\right]\]
}

\frame{
\frametitle{Classical EPA Equation}
\pause
求解感应流$\bar{J}^{sca}$和$\bar{M}^{sca}$的过程与传统矩量法类似，也是求解一个矩阵形式的线性方程组
\[\label{EPA:matrixequation}
\left[\begin{array}{cccc}
I & -S_{11}T_{12} & \dots & -S_{11}T_{1P}\\
-S_{22}T_{21} & I & \quad & -S_{22}T_{2P}\\
\vdots & \quad & \ddots & \vdots\\
-S_{11}T_{P1} & \dots & \quad & I
\end{array}\right]
\left[\begin{array}{c}
Q_1^{sca}\\
Q_2^{sca}\\
\vdots\\
Q_P^{sca}
\end{array}\right]
=\left[\begin{array}{c}
S_{11}Q_1^{inc}\\
S_{22}Q_2^{inc}\\
\vdots\\
S_{PP}Q_P^{inc}
\end{array}\right]\]
这里的$I$是单位阵，
\pause
\[Q_p^{sca}=\left[\begin{array}{c}
\bar{J}_p^{sca}\\
\bar{M}_p^{sca}
\end{array}\right]=
\left[\begin{array}{c}
\hat{n}_p\times\bar{H}^{sca}\\
-\hat{n}_p\times\bar{E}^{sca}
\end{array}\right]\]
和
\[Q_p^{inc}=\left[\begin{array}{c}
\bar{J}_p^{inc}\\
\bar{M}_p^{inc}
\end{array}\right]=
\left[\begin{array}{c}
\hat{n}_p\times\bar{H}^{inc}\\
-\hat{n}_p\times\bar{E}^{inc}
\end{array}\right]\]
为$ES_p\left(p=1,2,\dots,P\right)$上的入射和散射等效流强度。
}

\frame{
\frametitle{Advantages}
\pause
\begin{itemize}
\item 时间效率上：

表面积分方程法中有着精度和收敛速度难以调和的矛盾。
\pause
\begin{itemize}
  \item 矩阵的条件数与物体结构和剖分密度相关
\pause
  \item 复杂结构的物体表面流变化剧烈，为了精确求解需要密剖分
\end{itemize}
\pause
EPA多使用球形、柱形、长方体等结构做虚拟表面，这些结构的矩阵条件数较好。
\pause

\item 空间效率上：

虚拟面上的电磁流变化比较平缓，因而网格数通常远小于原物体表面的网格数。
\end{itemize}
}

\frame{
\frametitle{Improvement}
\pause
\begin{itemize}
\item Hybrid with PO-MM

\pause
耦合矩阵规模$N\times M$和$M\times N$，相乘的计算量为$N^2M$，应用EPA减少MM区未知量数目可进行平方加速，同时降低了内存的需求量。
\pause
\item Hybrid with DDM

\pause
通过观察
\[\left[\begin{array}{c}
\bar{J}_p^{sca}\\
\bar{M}_p^{sca}
\end{array}\right]=
S_{pp}\left[\begin{array}{c}
\bar{J}_p^{inc}\\
\bar{M}_p^{inc}
\end{array}\right]+
S_{pp}\left(\sum_{l=1,l\ne p}^PT_{pl}\left[\begin{array}{c}
\bar{J}_l^{sca}\\
\bar{M}_l^{sca}
\end{array}\right]\right)\]
的形式，发现其具有可并行化及区域分解的特性。
\end{itemize}
}

\frame{
\frametitle{Parallel-DDM based EPA}
\pause
\begin{figure}[htbp]
  \centering
  \includegraphics[scale=0.7]{../EPA_flow.eps}
\end{figure}
}

\frame{
\frametitle{EPA based Framework}
\pause
通过上面的论述，可以建立一种通用的基于EPA的快速算法框架
\pause
\begin{enumerate}
  \item 使用矩量法的区域都用虚拟等效面代替
\pause
  \item 统一化了外部求解的过程，不用根据矩量法区的材质而改变迭代方程
\pause
  \item 虚拟表面内部甚至可以不用矩量法，例如计算非均匀介质时可应用FDTD或FEM
\end{enumerate}
}

\frame{
\frametitle{Examples}
\pause
\begin{figure}[htbp]
  \includegraphics[scale=0.2]{../data/chapter5/cube.jpg}
  \includegraphics[scale=0.4]{../data/chapter5/sphere_cube.jpg}
\end{figure}
\pause
\begin{figure}[htbp]
  \includegraphics[scale=0.35]{../data/chapter5/cube_VV.pdf}
  \includegraphics[scale=0.35]{../data/chapter5/cube_HH.pdf}
\end{figure}
}

\frame{
\frametitle{Examples}
\pause
\begin{figure}[htbp]
  \includegraphics[scale=0.17]{../data/chapter5/2cube.jpg}
  \includegraphics[scale=0.17]{../data/chapter5/4cube.jpg}
  \includegraphics[scale=0.17]{../data/chapter5/8cube.jpg}
\end{figure}
\pause
\begin{figure}[htbp]
  \includegraphics[scale=0.3]{../data/chapter5/2cube_VV.pdf}
  \includegraphics[scale=0.3]{../data/chapter5/4cube_VV.pdf}
  \includegraphics[scale=0.3]{../data/chapter5/8cube_VV.pdf}
\end{figure}
}

\frame{
\frametitle{Examples}
\begin{figure}[htbp]
  \includegraphics[scale=0.4]{../data/chapter5/memory.pdf}
  \includegraphics[scale=0.4]{../data/chapter5/time.pdf}
\end{figure}
}

\frame{
\frametitle{Examples}
\pause
\begin{figure}[htbp]
  \includegraphics[scale=0.4]{../data/chapter5/SRR_array.pdf}
  \includegraphics[scale=0.3]{../data/chapter5/SRR_EPA_cube2.pdf}
\end{figure}
}

\frame{
\frametitle{Examples}
\begin{figure}[htbp]
  \includegraphics[scale=0.7]{../data/chapter5/SRR_6_6_3_EPA.pdf}
\end{figure}
}

\frame{
\frametitle{Examples}
\pause
\begin{figure}[htbp]
  \includegraphics[scale=0.29]{../data/chapter5/f16_EPA.jpg}
\end{figure}
}

\frame{
\frametitle{Examples}
\begin{figure}[htbp]
  \includegraphics[scale=0.4]{../data/chapter5/f16_EPA_PO_MM_VV.pdf}
  \includegraphics[scale=0.4]{../data/chapter5/f16_EPA_MLFMA_VV.pdf}
\end{figure}
\pause
\begin{table}[htbp]
  \caption{f16战斗机应用EPA前后的效率对比}
  \centering
  \begin{tabular}{lll}
    \hline
    计算方法 & 时间消耗(s) & 内存消耗(MByte)\\
    \hline
    使用EPA前 & 983.407 & 6000\\
    使用EPA后 & 92.52 & 900\\
    \hline
  \end{tabular}
  \label{tab:EPA_f16}
\end{table}
}

\frame{
\frametitle{Thank You!}
\pause
\textcolor{red}{\huge \quad 感谢参与答辩的诸位老师和同学！！}
}
\end{document}
